The normal distribution for points spreads

The question

"The line is home −4.5 at 1.95 each way. My ratings say the home side is 5.5 points better. What's that worth?"

How to price basketball and NFL handicap lines is a different problem from pricing football goals. Statometrics itself focuses on football. This lesson is here because the method, turning an expected margin into a probability with a bell curve, is the cleanest version of an idea every pricing model uses, and because it shows exactly why football needs something else.

The idea in one sentence

In high-scoring sports, the final margin spreads out roughly like a bell curve around the expected margin, so the chance of covering a line is the area of the curve beyond it.

The picture

In basketball a team scores 100 or more points from dozens of possessions. Add up lots of small, fairly independent scoring events and the total, and the margin between two teams, comes out close to a bell curve. That's the central limit theorem at work.

So the winning margin can be described with just two numbers:

• The expected margin, μ (mu). Your model's view of how much better one side is, for example home by 5.5.
• The spread, σ (sigma). How much real margins scatter around that. For NBA games it's roughly 12 points; for the NFL roughly 13 to 14. (Estimate your own from past errors.)

Draw a normal distribution with those two numbers. The handicap line is a vertical cut. The area to the right of the cut is the chance the favourite covers.

Home handicap line	Chance home covers (μ = 5.5, σ = 12)
−2.5	59.9%
−3.5	56.6%
−4.5	53.3%
−5.5	50.0%
−6.5	46.7%
−7.5	43.4%

Each point of line near the middle is worth about 3.3% of probability. Half a point is worth fighting for.

Why this doesn't work for football

Football margins are small whole numbers: most matches finish somewhere between −2 and +2, with a big spike at 0 for draws. A smooth bell curve can't capture that. For football you price the goal difference from the scoreline grid in Lesson 4.2, or from the Skellam distribution, which is the difference of two Poisson counts.

Worked Betfair example

An NBA game. Your ratings make the home side 5.5 points better, with a spread of 12. Betfair's handicap market has home −4.5 at 1.95. (Illustrative figures.)

1. How far is the line from your view? You expect +5.5 and the home side needs +4.5 or more (5 or more, since there are no half points). The gap is 5.5 − 4.5 = 1.0 point.
2. Turn the gap into standard deviations. z = 1.0 ÷ 12 = 0.083.
3. Area beyond the line. The chance of a normal result above −0.083 standard deviations is 53.32%. That's the chance home covers.
4. Fair price. 1 ÷ 0.5332 = 1.875.
5. Break-even after 2% commission. About 1.893. Betfair's 1.95 is above that.
6. EV on £20 at 1.95. Win 0.5332 × £19 × 0.98 = £9.93. Lose 0.4668 × £20 = £9.34. EV = +£0.59, about +3p per £1.
7. The moneyline from the same curve. Home win = area above 0 = Φ(5.5 ÷ 12) = 67.66%, a fair price of 1.478.
8. Totals work the same way. Expected total 224, spread 18, line 219.5: chance of over = 59.87%, fair 1.670.

Verdict: a one-point disagreement with the market is worth about 3% of probability, which after commission is a thin edge. In a sharp market, a one-point gap is more often your model's error than the market's. Test it against closing line value before trusting it.

The formula

Chance of covering

$$P(\text{cover}) = 1 - \Phi\!\left(\frac{L - \mu}{\sigma}\right) = \Phi\!\left(\frac{\mu - L}{\sigma}\right)$$

• Φ (phi) is the area under a standard bell curve to the left of a point.
• L is the margin needed to cover (4.5 for a −4.5 line).
• μ is your expected margin.
• σ is the spread of real margins around your expectation.

In plain English: measure how far your expected margin is past the line, in units of the spread, and read the area off the bell curve.

The moneyline

$$P(\text{win}) = \Phi\!\left(\frac{\mu}{\sigma}\right)$$

In plain English: a straight win is just a handicap line of 0.

Where σ comes from

$$\sigma \approx \sqrt{\frac{1}{n}\sum_{k=1}^{n}\big(\text{actual margin}_k - \text{predicted margin}_k\big)^2}$$

• n is the number of past games you've predicted.

In plain English: σ is the typical size of your model's misses. Measure it from your own predictions, not someone else's.

Try it

Pen and paper: same game, but now the line moves to home −6.5. What's the chance home covers, and would 1.95 still be a bet?

Answer

z = (5.5 − 6.5) ÷ 12 = −0.083, so the chance is 46.68%. The fair price is about 2.14 and the break-even after 2% commission about 2.17. Backing home at 1.95 would be a losing bet; the other side of the line would now interest you.

Common mistakes

• Using the normal curve for football. Goal margins are small and lumpy with lots of draws. Use a Poisson grid or Skellam instead.
• Ignoring key numbers. NFL margins bunch on 3 and 7 because of how points are scored. A smooth bell curve under-rates those margins, so lines crossing 3 or 7 need special care.
• Borrowing someone else's σ. The spread is the size of your model's misses. A worse model has a wider σ and should be less confident.
• Treating a one-point edge as a big one. One point is about 3% of probability, which commission and model error eat quickly.
• Forgetting that the market is also a model. A handicap line is the market's expected margin. If you disagree, the burden of proof is on you.

Overconfidence in the width of your own errors is one of the quiet reasons good models still lose money.